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Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost

  • Tomasz Miller
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Abstract

We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.

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Cite

BibTeXRIS
BibTeX
@misc{miller2026metricity,
  title = {{Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost}},
  author = {Tomasz Miller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.03769},
  eprint = {2609.03769},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03769},
}
RIS
TY  - UNPB
TI  - Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost
AU  - Tomasz Miller
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.03769
UR  - https://arxiv.org/abs/2609.03769
AB  - We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.
ER  -